Download Algorithmic Game Theory Fall 2016-2017 Exercises 3 15. Consider

Algorithmic Game Theory
Fall 2016-2017
Exercises 3
15. Consider the local diameter network creation game in which we have
a set V of n. A player can select strategically its set of neighbors.
For a given profile s = (s1 , . . . , sn ), let G[s] be the undirected graph
induced by the player’s decisions. The cost for player u ∈ V is defined
by means of three parameters α, β, w ≥ 0 according to the following
expression:
cu (s) = α|su | + w #{v | dG[s] (u, v) > β}.
Analyze under which parameter assignments the following graphs are
NE.
• Sn a star graph.
• Kn a complete graph.
• In an independen set.
• A tree with diameter β.
16. Assume that we have a graph with edge and node non-negative weights,
i.e. G = (V, E, w, b) where w : E → Z+ and b : V → Z+ . An information gathering game is defined on top of G as Γ = (I, G) where I ⊆ V .
The game has m = |I| players. Player i ∈ I can select any path in
G starting at node i. The players use the selected path to gather the
information hidden at the nodes of the graph. In order to traverse a
path they have to pay the toll fees represented by the edge weights.
The value of the information hidden in a node u is b(u). However,
the value of the information degrades proportionally to the number of
players that discover such a piece of information.
• Provide a formal definition of the cost function for the information
gathering game.
• Does this family of games have always a PNE?
17. Consider a set of n players that must be partitioned into two groups.
However, there is a set of bad pairings and the two players in such a
pair do not want to be in the same group. Moreover, each player is
free to choose which of the two groups to be in. We can model this by
a graph G = (V, E) where each player i is a vertex. There is an edge
(i, j) if i and j form a bad pair. The social objective is to maximize
the number of edges between the two groups. The private objective
of player i is to maximize the number of its neighbors that are in the
other group.
• Show by a potential function argument that this game has a Pure
Nash equilibrium.
• Show that the PoS is 1.
18. The owner of an item is planning to increase his benefit by being one
of the bidders in the selling auction. The auction will take the format
of an English auction. If the owner wins, he will not pay anything
and the item remains his property. Assume that there are n bidders
(other than the owner) and that the value of the item for each bidder
is drawn uniformly at random from [0, d]. Also the value of the good
for the owner is v. The auction bids increases by a value e which is
vanishingly small compare to d. Can the owner get a higher benefit
by participating? If so, what is the owners optimal bidding strategy?
19. Consider a combinatorial auction in which we have n bidders and a set
O of m objects for sale. Each bidder has a valuation vi (S), for S ⊆ O.
The aim of the seller is to maximize revenue.
• Assuming that O = {a, b, c} and that there are three players
{A, B, C}. The valuations each player has for each subset of the
items is
A
B
C
a
1
3
4
b
2
5
5
c
1
3
0
ab
6
5
7
bc
4
5
5
ac
5
3
7
abc
11
5
9
What is the allocation produced by the VCG mechanism? What
are the VCG payments?
• Assuming that the bids are truthful give an integer program formulation to compute the VCG allocation.
20. Consider an auction in which the seller announces a reserve price of r
before running the auction. With a reserve price, the item is sold to
the highest bidder if the highest bid is above r; otherwise, the item
is not sold. In a first-price auction with a reserve price, the winning
bidder (if there is one) still pays her bid. In a second-price auction with
a reserve price, the winning bidder (if there is one) pays the maximum
of the second-place bid and the reserve price r.
• Is truthful bidding a dominant strategy on such auctions?
• Assuming that the seller assigns value u to the object and chooses
the reserve price r in order to maximize its expected revenue in a
SP auction. Show that r might be higher than u. (Hint. Provide
a value for the simple case in which the item is worth u = 0 to
the seller and a second-price auction with a single bidder, whose
value is uniformly distributed on [0, 1] is run.)
21. Consider a GSP auction for n players.
• Show that any bid bi > vi is dominated by bid bi = vi .
• Show that every envy-free equilibrium is efficient.
• Show that the PoS of the GSP mechanism is 1.
22. Consider a GSP auction on n bidders in which all the valuation factors
are 1. We say that a bid profile b is up-Nash for player i if he can’t
increase his utility by taking some slot above the one in the corresponding allocation π. A bid profile is up-Nash if it is up-Nash for all
players.
Show that if a bid profile b is a Nash equilibrium, then the bid profile
b0i = bπ(i) is up-Nash.
23. Consider a GSP auction on n bidders in which all the valuation factors
are 1 (γi = 1) and the corresponding VCG mechanism.
• Show that GSP can generate more revenue than VCG and viceversa.
• Show that the GSP revenue in a Nash equilibrium is always at
least half of the revenue of the VCG mechanism on the auction
in which the participants are all the bidders except the one with
the highest valuation. (Hint. Use the construction on problem
22)